calculating shaft critical speed by hand 1

[hector002]

I am trying to calculate(estimate) the critical shaft speed of a rotating shaft.

When i did this back in school we solved for the eiganvalues of the 4th order diff equation:

d^4/dx^4 - B^4*y=0 where B^4=rho*A/(E*I)*omega

however this assumes A (shaft area) to be constant. The shaft i an interested in steps from 1" to 2" and back down to 1" between the bearings. Does anyone have some tips or references to solve eigenvalue problems for a shaft with non uniform diameter. thanks

 

[ivymike]

just calculate an approximate bending stiffness of the shaft (probably don't account for the entire "benefit" of the increased diameter lobes, and if this looks like a camshaft, remember that you'll have to consider shear as well as bending when you estimate the deflection, because it's not a long & slender beam), use the deflections under a unit load (at local CG) to come up with an equivalent mass of the shaft (referred to the local CG), and use the stiffness with the mass in a 1-DOF system to estimate the first natural frequency. If I remember correctly, that'll be pretty close to your whirl frequency.

 

[GregLocock]

I don't think there is a closed form equation for it. So, you can either work it out from calculus - you sound as though you may be comfortable with that approach, use FEA, or set up a spreadsheet and use rayleigh ritz.

Blevins has a formula for the frequency of a massless beam with a concentrated mass on it (critical frequency of a shaft is the same as its bending frequency) which may help

2/pi*sqrt(3EI/(M*L^3))



Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[electricpete]

A twist of Greg's last solution allows mass in both the beam and the concentrated mass.

Mark's handbook 8th edition page 5-70:

Beam of length l and mass mb simply supported at both ends with mass m in the middle:

wn = sqrt(48*E*I /[(m+0.5mb)*l^3])

A quick comment on iterative methods. I have never solved any yet but I am getting ready to. "Handheld Calculator Programs For Rotating Equipment Design" by Fielding is available used for $10 including shipping. In spite of the title it was geared for the days when programmable calculators were widely available and computers weren't... so the algorithms are suitable for computers. Chapter 1 has a finite element technique called Prohl Myklestad. It looks fairly straightforward. You have to iterate first to find the critical frequencies (the frequencies which satisfy the boundary conditions and make some residual = 0). Then you can easily compute the mode shapes once critical frequencies are known. Some advantages over Raleigh are that you don't need to guess the mode shape, you can calculate higher order critical speeds without knowing the mode shape, and you can calculate the mode shape.

If you could wait about two months I hope to have it programmed in excel and vba.

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[electricpete]

Let me use L instead of l for readability:


Beam of length L and mass mb simply supported at both ends with mass m in the middle:

wn = sqrt(48*E*I /[(m+0.5mb)*L^3])


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[electricpete]

and of course fn = wn/(2*Pi)

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[GregLocock]

http://www.findarticles.com/p/articles/mi_qa4075/is_200306/ai_n9296359

Transfer matrix method - now this almost sounds like the theory of receptances, which is a hand powered method for predicting the dynamic response you get when you join two dynamic systems.



Cheers

Greg Locock

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[hector002]

thanks everyone for the help. I failed to mention that the shaft is 2" for the majority of the length and only drops down to 1" at the bearings so Im worried a concentrated mass might be too rough an estimate (but a good idea-ill probably use that to check the reasonableness of my answer.

Yeah, i was hoping for a nice closed solution but with the discontinuities of the diameter along the lenght it looks unlikely. My only thought was that you could make the Area a function of length using a step or dirac delta function and solve. But that is pushing the limits of my math skills. The next best bet is to start brushing up on my numerical methods. Thanks

 

[electricpete]

Unfortunately I think you are coming to the correct conclusion that numerical methods are required for all but the simplest geometries.

Greg's comment makes me think that transfer matrix is a more appropriate term to describe the algorithm I mentioned (Leslie's book) than finite element. You break the shaft into sections and relate the conditions at one end of shaft to other end of shaft via a matrix transformation of the 4 variables ... something like y, displacement, angle, shear, moment.

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[GregLocock]

How many bearings are there? what sort? how long are they?



Cheers

Greg Locock

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[hector002]

elecpete- what is the title of leslie's book you are referencing?

greg, im coming clean- the dim i gave were just hypothetical because i was too lazy to look at the drawings.

The actual specs are:

4" dia x 9.5" section centered between 50mm (give or take due to bearing stops on shaft) sections

im using two pairs of 210 dulex angular bearings (40mm wide for the two rows) spaced 11.5" apart.

My plan was to model the shaft fixed-fixed.

 

[GregLocock]

I bet it is closer to pinned with those dimensions. You can try to get an estimate of the bearing stiffness from the manufacturer to confirm that.

So I think the problem collapses down to the simplest possible - a pinned uniform shaft.

Are you going to get any test work done?

Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[hector002]

If you dont mind, could you elaborate on why you think a pinned uniform shaft would be a good approx. My practical experience with shafts and critical speeds is almost zero but I was thinking that it seems a 40mm(~1.5") wide double row bearing would be signifigently large in rigidly clamping a 12" long shaft. And it seem that bending wise that that 4" section would be very stiff and the majority of the bending would occur in the 50mm sections.

Also, what sort of test work were you referring to?

 

[GregLocock]

My experience with bearings is that they are not as rigid as you'd hope! say of the order of 10000 Nm/degree (details, especially preloads, are important in working this number out). 10000 Nm/degree may sound stiff, but in the context of a wheel bearing, for example, that can give quite measurable deflections in service.

"it seem that bending wise that that 4" section would be very stiff and the majority of the bending would occur in the 50mm sections." exactly, so the ends would be like pin joints, even if the bearings are rigid, which in the context of a 4 inch shaft is unlikely...

as to test work, run it through the speed range to find where the critical speeds are, or hit it with a modal hammer to measure the bending frequencies, or both.







Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[electricpete]

Hector - my bad. The author was not Leslie. It was Leslie Fielding.

TITLE: Handheld Calculator Programs for Rotating Equipment Design
by Leslie Fielding
ISBN: 00702-0695-3
Publisher: McGraw-Hill
Publish Date: February, 1983
Binding: Hardcover
List Price: USD 52.95

If this links works correctly you will see it begins at $10 for used:

http://www.addall.com/New/BrowseCom...ulator+Programs+for+Rotating+Equipment+Design

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[ivymike]

as I mentioned earlier - you probably shouldn't exclude shear deflection from your estimate of stiffness if the shaft is so thick. Also, if the section is stepped, don't expect the ends of the thick parts to resist bending the way they would if the whole shaft were that thick.

With a shaft that is almost all 4" wide, how likely is it that you'll spin it up to anything near its critical frequency?

Two quick checks: If it were pinned at the ends, and had a uniform 50mm section, would you get close to the critical speed? What if you had a uniform 100mm cross section? If the answer to the first check is "no," then why worry about the real shaft? If the answer to the second check is "yes," then you know you'll have a problem with the stepped shaft, right?

 

[rob768]

To make things more difficult: are there disk or rotors on the shaft, so a whirling effect may be expected?

 

[hector002]

rob, no there are no rotors or disks but as this shaft is going to be used for a fluid bearing test rig i will be concerned with oil whip/whirl at 1/2 the critical speed.

greg, no physical tests yet as i am still in the design stages of all this.

((what i am trying to go is find a maximum operating speed for my rig so i can open up the radial clearances of my fluid bearing-so when i measure displacements runout and other background noises will comparatively smaller to increase SN ratio)).

ivymike, i like te idea of the quick check- i think something along these lines will be good enough for my purposes. However is that sound logic that if a 50mm uniform shaft is OK that my 50,4",50 shaft will be OK too, i would guess that adding mass in the middle would lower the critical speed, but maybe not since the 4" section runs almost the majority of the length.

 

[electricpete]

http://home.houston.rr.com/electricpete/engtipsrot4.doc

Above is output of a critical speed/mode shape program that uses the algorithm I mentioned above.

Of course the output is only as good as the input.

Check the first figure to see if the geometry is right.

Mode shapes are shown in the 2nd figure. 1st critical at 68000cpm and 2nd crit above 100kcpm.

Below that is program output which describes the options I used in running the program. See if you agree.

I think I got carried away cranking up bearing stiffness. At lower stiffnesses I was getting a very flat looking first critical mode shape, so I cranked up the stiffness to get a pretty bowed shape. But now that I think about it, this rotor without disks may not deform much at first critical if most of deflection is occuring in the bearings.

If you have some changes to suggest, I will be glad to rerun the program.

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[electricpete]

"fluid bearing test rig i will be concerned with oil whip/whirl at 1/2 the critical speed."

I'm not an expert on this subject. Oil whirl of course occurs at a frequency slightly below half running speed. Wouldn't oil whip then occur when running speed slightly above twice critical speed so that the whirl frequency is near the rotor critical frequency?

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